Generalized Stirling permutations, families of increasing trees and urn models

TL;DR

This paper extends the study of Stirling permutations by exploring their generalizations, connecting them with various families of increasing trees and urn models, and analyzing their asymptotic behaviors.

Contribution

It introduces generalized Stirling permutations, relates them to new families of increasing trees, and analyzes their parameters using urn models and bijections.

Findings

Established connections between generalized Stirling permutations and increasing trees.

Derived asymptotic distributions of parameters using urn models.

Provided bijections between different classes of increasing trees.

Abstract

Bona [2007+] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [1978]. Recently, Janson [2008+] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of Bona and Janson, and relate them with certain families of generalized plane recursive trees, and also $(k+1)$-ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) Polya urn models using various methods.